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From Metropolis to diffusions: Gibbs states and optimal scaling. (English) Zbl 1047.60065

Let Π=(π W (a,dx),W d finite,a d ) be a specification, that is, a consistent family of finite-volume conditional Gibbs measures for a finite-range Hamiltonian H. Suppose that ξ is the corresponding infinite-volume Gibbs measure, let ξ be translation invariant. Given a subset V n d of cardinality n and a boundary condition z, let (X t (V n ,z),t0) be the random walk Metropolis chain for π V n (z,·). It was shown by the second author and A. F. M. Smith [Stochastic Processes Appl. 49, 207-216 (1994; Zbl 0803.60067)] that X t (V n ,z) converges weakly to π V n (z,·) as t. The behaviour of the algorithm as V n d is studied. In particular, choose the proposal variance σ n 2 =ln -1 . Under suitable assumptions on H and ξ it is proven that X [nt] (V n ,z) converges weakly as n to an infinite-dimensional diffusion Z t on a Hilbert space E=L 2 ( d ,ρ). The measure ρ is given by ρ({k})=( j d exp(-|j|)) -1 exp(-|k|), Z solves the equation

dZ t =-l 2v(Z t )H(Z t )dt+lv(Z t )dB t ,

driven by a Brownian motion B in E and with the initial condition Z 0 =ξ in law. The coefficient v is defined in terms of the second derivative of the Hamiltonian H.

MSC:
60J05Discrete-time Markov processes on general state spaces
65C05Monte Carlo methods
60J22Computational methods in Markov chains